{
  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $f \\colon U \\to \\mathbb{R}^n$ be of class $C^1$ with $Df(x)$ invertible for each $x \\in U$.\nThen each $a \\in U$ has an open neighbourhood $V \\subseteq U$ such that $f(V)$ is open and $f$ restricted to $V$ is a homeomorphism onto $f(V)$.\nIn particular, $f$ is an open map: $f(O)$ is open for each open set $O \\subseteq U$.\n\n## Remarks\n\nThis follows from the [[inverse function theorem|0K9HSJFCMJ]], since a $C^1$ inverse is [[continuous|0S9EK5MQPE]].\nFor each $a$, the hypothesis is needed only at $a$: if $Df(a)$ is invertible, $f$ is a homeomorphism from a neighbourhood of $a$ onto an open set.\nA local homeomorphism need not be injective. The map $(x, y) \\mapsto (e^x \\cos y, e^x \\sin y)$ is a local homeomorphism from $\\mathbb{R}^2$ onto $\\mathbb{R}^2 \\setminus \\{0\\}$, but it is not injective.\n\n## Explanations\n\n### L1 human\n\nWhere the derivative is invertible, the map neither folds nor crushes space in a small region.\nSo small open sets go to open sets, and the map can be undone continuously on each small region.\n",
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    "name": "Local homeomorphism from the inverse function theorem",
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  "msc": [
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  "name": "Local homeomorphism from the inverse function theorem",
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